### Signals with pitch characteristic trend

Trends with pitch characteristics in the signals, which are collected during an impact vibration test by piezoelectric acceleration sensors, are the result of a combination of different factors. Zhang et al.1 analyzed the reasons for the trend appearing under high impulse and high overload conditions by the acceleration sensors. Figure 1 shows the accelerations collected during a gun test at different measurement points. The accelerations of points 1 and 2 in the z-direction have characteristic stepped trends, which are illustrated in Figures 1c,f.

### Extraction of trends by LFE method

Common trend extraction methods, such as the five-point triple smoothing method, the wavelet transform method, and the method based on the empirical mode decomposition, have certain limitations in extracting the characteristic trend from the step in Figure 1, which will be discussed later. This article proposes a method using a logistic function and upper and lower envelopes to extract the characteristic trend of the stage. The method consists of the following steps:

Step 1: Determine the starting position of the staircase trend by a logistic function.

Many cases of trends occurring under high impulse and high overload conditions are shown in Fig. 1c, f. According to experience, when the baseline of the signal drifts, the trend is usually a sudden drop, which looks like an inverted S-shaped curve. Therefore, we select an inverted S-shaped function and calculate the mutual correlation between the S-shaped function and the signal there(you). When the S-shaped function is relative to the signal with a characteristic pitch trend, the slope k mutual correlation curves is the maximum. We can effectively determine the starting position of the staircase trend in this way. The logistic function is a common S-shaped function:

$$Pleft( t right) = frac{{KP_{0} e^{rt} }}{{K + P_{0} left( {e^{rt} – 1} right)} }$$

(1)

or (P_{0}) is the minimum value of the curve, (K) is the maximum value of the curve, and (r) is the logistic growth rate or the slope of the curve.

Step 2: Identify all local extremes (maxima and minima) of there‘(you1), which consists of the signals from the start position of the step characteristics trend to the end. Meanwhile, set the rest of the signals there(you) like there‘(you0).

Step 3: Connect every two neighboring local maxima (minima) by a linear equation to determine the upper envelope Emaximum(you1) (the lower envelope Emin(you1)).

Step 4: Construct the Average of the Empirically Determined Upper and Lower Envelopes m

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